• Approaches to Quantum Control and Quantum Engineering

Thomas Schulte-Herbrüggen (Technische Universität München and Munich Centre for Quantum Science and Technology)

Fundamental questions in quantum systems theory and control engineering include: to which extent is a quantum control system

• controllable, or 

• accessible, or 

• observable, or

• stabilizable ? 

Likewise, given the controlled equation of motion (for closed or open Markovian systems) plus an initial condition: what is  

• the reachable set of quantum states, or (on the operator lift)

• the set of reachable quantum maps ?

Finally, what can one do with Markovian (thermal) resources and where is the border to non-Markovian control systems?

The tutorial sets out to sketch a unified Lie-theoretical frame for addressing (if not answering) these questions rigorously. It is meant as a step towards a dedicated quantum systems theory for the quantum engineer.

We introduce and exemplify system-theoretical principles and tools by analysing systems following an extended Lindblad master equation. They all take the form of (bilinear) control systems, a form which sets the unifying Lie-frame frame for the tutorial. In this picture, a Quantum Noether-type Theorem naturally arises. Not only does it relate symmetries to fixed points, but it allows for symmetry assessment of many of the questions above.

By recent examples we illustrate how breaking system symmetries by (optimal) controls then paves the way to exploiting the full quantum potential of experimental set-ups paradigmatic or pertinent in quantum engineering and emerging technologies.

• Extensions of quantum mechanics canonically emerging from the combination of classical probability with the theory of orthogonal polynomials

Luigi Accardi (Università di Roma Tor Vergata - Centro Vito Volterra)

Since the first steps of quantum probability (QP) in the years 1970s, it was a folklore statement that QP is a generalization of classical probability (CP). Developments over the last 25 years have shown that this statement must be completely overturned, namely: the whole quantum theory (QT) can  be deduced from the combination of CP with the classical theory of orthogonal polynomials (OP). The importance of the new point of view does not lie so much in the above-mentioned deduction, but in the fact that, in this deduction, the known quantum theory appears as a very special case, corresponding to some specific classical probabilistic requirements, of a much broader deductive theory, which allows to take into account other, often more realistic, probabilistic requirements. In other words, for the first time in over 100 years, the mathematical apparatus of QT appears in the perspective of a natural deduction and not as a strange, singular theory justified a posteriori by its enormous empirical success, but totally mysterious in its origins and meaning. The fall-outs of this new point of view, both for QT and for CP have already been very large, but this is only the beginning. Some of these fall--out will be discussed in the course. It is not hazardous to expect that, in the near future, both these disciplines will undergo a radical conceptual and technical innovation.

Goal of the present course is to explain, in a clear and concise form, the new ideas and techniques as well as the open problems emerging from this approach. The starting point of the theory is the remark that any random variable X with all moments can be expressed as the sum of 3 operators called Creation, Annihilation and Preservation (CAP) operators. This is called the canonical quantum decomposition of X. These operators can be characterized by a set of commutation relations (CR) which are natural extensions of Heisenberg commutation relations. The latter characterize the Gaussian class. In turn, the CR algebraically characterize the classical random variable X in the sense of moment equivalence. Fermions are included through an algebraic extension of the notion of 'stochastic coupling' applied to classical Bernoulli random variables. More generally, every random variable with all moments defines its own extension of usual quantum mechanics, including momentum operator, second quantization, normal order, ... The theory is valid in any dimension, finite or infinite. In particular (non-relativistic) boson quantum field theory is deduced from the standard gaussian process on a Hilbert space. To follow the course it is sufficient to be familiar with the classical probabilistic terminology and with elementary properties of Hilbert spaces.

• Completely positive maps and evolutions

B. V. Rajarama Bhat (Indian Statistical Institute Bangalore, Bengaluru, India)

Lecture 1: Completely positive maps, Stinespring's theorem and Choi-Kraus representation

Lecture 2: Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) Theorem

• Quantum Properties of Classical Pearson Random Variables

Un Cig Ji (Chungbuk National University, Cheongju, South Korea)

In this talk, we discuss the properties of the canonical quantum decomposition of the classical random variables, specially in the Pearson class. We focus on the problem of representing the creation-annihilation-preservation (CAP) operators canonically associated to a real valued random variable with all moments as (normally ordered) differential operators with polynomial coefficients. We deduce explicit formulas for the polynomial coefficients in the representation of the CAP operators. We give a new characterization of the Pearson distributions in terms of the hermitianity of the associated Sturm-Liouville operators. We also introduce the notion of finite type random variable and then characterize type-2 and type-2 real-valued random variables.

We discuss a necessary condition, for a real random variable to be of finite type, which is the polynomial growth of the corresponding principal Jacobi sequence.

This allows to single out three classes of random variables of infinite type and to prove that the Beta and the uniform distributions are of infinite type.

This talk is based on a joint work with L. Accardi, A. Ebang Ella and Y. G. Lu.

• Superoperator Master Equations and Effective Dynamics

Alexander E. Terentekov (Steklov Mathematical Institute of RAS, Moscow, Russian Federation)

I will present some superoperator analogs of quantum master equations. The similarities and differences between the derivations of the usual and the superoperator master equations will be discussed. I will also talk about the physical motivation for considering such equations. Then I will give some particular examples of superoperator master equations that are closely related to dynamics with effective Hamiltonian.

Registration to the school is free, but mandatory and needs to be done by April the 7th (in person attendance limited due to room capacity).

Registration link: coming soon

Supported by the Project Dipartimenti di Eccellenza 2023-2027 (CUP D43C22005050006)